The Truncated Tetrahedral Cupoliprism

The truncated tetrahedral cupoliprism, also variously known as the parabidiminished rectified tesseract, truncated tetrahedron atop inverted truncated tetrahedron, the Bowers acronym tutcup, or K4.55 in Klitzing's list of segmentochora, is a CRF polychoron formed from the convex hull of two truncated tetrahedra placed in dual orientation to each other in parallel hyperplanes, with 6 tetrahedra and 8 triangular cupolae connecting them. It has a total of 16 cells, 52 polygons (32 triangles, 12 squares, 8 hexagons), 60 edges, and 24 vertices.

The truncated
tetrahedral cupoliprism

Even though it is not uniform because of the triangular cupola cells, it is nonetheless vertex-transitive, and thus belongs in the category of scaliform polytopes.

Structure

We shall explore the structure of tutcup using its parallel projections into 4D, centered on a truncated tetrahedral cell.

Parallel projection of
the truncated tetrahedral cupoliprism, showing nearest truncated
tetrahedron

The above image shows the nearest truncated tetrahedron to the 4D viewpoint. For clarity, we have left out the other cells.

The hexagonal faces of this truncated tetrahedron are joined to 4 triangular cupolae, shown next:

Parallel projection of
the truncated tetrahedral cupoliprism, showing 1/4 triangular cupolae

Parallel projection of
the truncated tetrahedral cupoliprism, showing 2/4 triangular cupolae

Parallel projection of
the truncated tetrahedral cupoliprism, showing 3/4 triangular cupolae

Parallel projection of
the truncated tetrahedral cupoliprism, showing 4/4 triangular cupolae

Here are all the cells together:

Parallel projection of
the truncated tetrahedral cupoliprism, showing nearest truncated tetrahedron
and 4 triangular cupolae

Where the triangular faces of the cupolae touch each other, are 6 tetrahedral cells:

Parallel projection of
the truncated tetrahedral cupoliprism, showing 6 equatorial tetrahedra

These tetrahedra have been foreshortened into squares because they lie at a 90° angle to the 4D viewpoint. But in reality, they are perfectly regular tetrahedra. For clarity, we've left the cells on the near side of the polychoron only in edge outline.

The far side of these tetrahedra are joined to 4 more triangular cupolae on the far side of the polytope:

Parallel projection of
the truncated tetrahedral cupoliprism, showing 1/4 far side triangular
cupolae

Parallel projection of
the truncated tetrahedral cupoliprism, showing 2/4 far side triangular
cupolae

Parallel projection of
the truncated tetrahedral cupoliprism, showing 3/4 far side triangular
cupolae

Parallel projection of
the truncated tetrahedral cupoliprism, showing 4/4 far side triangular
cupolae

The cavity surrounded by these cupolae are filled by the antipodal truncated tetrahedron:

Parallel projection of
the truncated tetrahedral cupoliprism, showing antipodal truncated
tetrahedron

Here are all the far side cells together:

Parallel projection of
the truncated tetrahedral cupoliprism, showing all far side cells

The following table summarizes the cell counts for the truncated tetrahedral cupoliprism:

Region
Near side	1	0	4
Equator	0	6	0
Far side	1	0	4
Grand total	2	6	8
Grand total	16 cells

Coordinates

The Cartesian coordinates of tutcup (K4.55), centered on the origin with edge length 2, are:

(±1, −1/√3, 5/√6, 1/√2)
(0, 2/√3, 5/√6, 1/√2)
(±2, −2/√3, 1/√6, 1/√2)
(0, 4/√3, 1/√6, 1/√2)
(±1, ±3/√3, −3/√6, 1/√2)
(±2, 0, −3/√6, 1/√2)
(±1, ±3/√3, 3/√6, −1/√2)
(±2, 0, 3/√6, −1/√2)
(±2, 2/√3, −1/√6, −1/√2)
(0, −4/√3, −1/√6, −1/√2)
(±1, 1/√3, −5/√6, −1/√2)
(0, −2/√3, −5/√6, −1/√2)

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